Mermin–Wagner

A no-go theorem with a sharp, checkable statement: a system with a continuous internal symmetry and short-range interactions cannot spontaneously break that symmetry at any finite temperature in dimension $d\le 2$. For magnetism the corollary is the one that bites: the isotropic Heisenberg model in one or two dimensions has zero spontaneous magnetisation for every $T>0$. The would-be order is destroyed not by some large fluctuation but by the cheapest ones — the long-wavelength Goldstone modes (here, magnons), whose population diverges in the infrared in low dimensions. The theorem is the reason “why is there 2D magnetism at all?” is a real question, and the answer — anisotropy, the order-restoring ingredient that connects directly to DMI & anisotropy — is what makes the upstream DFT have to capture more than the isotropic $J$.

Prerequisites

the long-wavelength Goldstone magnon dispersion $\omega (\mathbf{k})\sim D{k}^2$ at small $\mathbf{k}$ — the elementary consequence of the finite second moment of the short-range exchange (no Holstein–Primakoff or Colpa machinery needed) · the Heisenberg / spin model $H=-{\sum }_{ij}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ (plain text — no dedicated note on disk) · the classical/quantum partition function and equilibrium correlation functions

The precise statement

Take the Heisenberg Hamiltonian with finite-range, finite-strength exchange,

$$H=-\sum _{i\ne j}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j-\mathbf{B}\cdot \sum _i{\mathbf{S}}_i,\sum _j|J_{ij}||{\mathbf{R}}_i-{\mathbf{R}}_j{|}^2<\infty ,$$

on a $d$-dimensional lattice. The continuous symmetry is global spin rotation $O(3)$ (or $O(2)$ for the $XY$ model), broken by the symmetry-breaking field $\mathbf{B}$. Mermin and Wagner (1966) proved that for $d\le 2$ the magnetisation obeys a bound of the form

$$|⟨\mathbf{S}{⟩}_{\mathbf{B}}|\le \{\begin{array}{ll}\frac{\text{const}}{\sqrt{T}\sqrt{|\ln |\mathbf{B}||}},& d=2,\\ \text{const}\times (\frac{|\mathbf{B}|}{T^2}{)}^{1/3}=\text{const}\times \frac{|\mathbf{B}{|}^{1/3}}{T^{2/3}},& d=1,\end{array}$$

so that the spontaneous magnetisation, obtained by taking $\mathbf{B}\to 0$ after the thermodynamic limit, vanishes for any $T>0$. The second moment condition on $J_{ij}$ is the precise meaning of “short-range.” The result is non-perturbative — it bounds the exact order parameter — and holds for the quantum model at any spin $S$. Hohenberg (1967) had proved the analogous statement for superfluid/crystalline order, and the magnetic case is sometimes called Mermin–Wagner–Hohenberg.

Proof sketch: Bogoliubov inequality + the IR-divergent fluctuation integral

The engine is the Bogoliubov inequality, a rigorous thermodynamic bound relating a thermal average to a susceptibility-like double commutator. For any pair of operators $A,C$,

$$\frac{1}{2}⟨\{A,A^{†}\}⟩\cdot ⟨[C^{†},[H,C]]⟩\ge k_{B}T|⟨[C,A]⟩{|}^2.$$

Read it as: the size of a fluctuation ($⟨\{A,A^{†}\}⟩$) is bounded below by an order parameter ($|⟨[C,A]⟩{|}^2$) divided by an energy cost (the double-commutator). Now choose the operators to probe magnetisation at wavevector $\mathbf{k}$: let $A=S^{-}(-\mathbf{k}-\mathbf{K})$ and $C=S^{+}(\mathbf{k})$, the spin-density Fourier components, with $\mathbf{K}$ a reciprocal-lattice vector of the assumed magnetic order. Then

• the commutator $⟨[C,A]⟩$ is proportional to the magnetisation $M=⟨S^z⟩$ — this is the order parameter we want to bound;
• the double commutator $⟨[C^{†},[H,C]]⟩$ is bounded above by the energy stored in a spin-wave of wavevector $\mathbf{k}$, which for small $\mathbf{k}$ goes as $$⟨[C^{†},[H,C]]⟩\lesssim |\mathbf{B}|M+\text{const}{k}^2,$$ the field supplying a Zeeman gap $\propto |\mathbf{B}|M$ and the $k^2$ being the exchange stiffness of the Goldstone mode (short-range $J$ with a finite second moment is exactly what makes the leading small-$\mathbf{k}$ behaviour analytic and $\propto k^2$; the full magnon spectrum is developed in linear spin-wave theory).

Rearranging the inequality bounds the transverse fluctuation from below by $M^2/(|\mathbf{B}|M+\text{const}{k}^2)$, and summing the sum rule ${\sum }_{\mathbf{k}}⟨\{A,A^{†}\}⟩=NS(S+1)$ over the Brillouin zone gives, in the continuum limit,

$$S(S+1)\ge \text{const}\times M^2{k}_{B}T{\int }_{\mathrm{BZ}}\frac{d^{d}k}{(2\pi {)}^d}\frac{1}{|\mathbf{B}|M+\text{const}{k}^2}.$$

Everything hinges on this integral as $B\to 0$. The angular part contributes $k^{d-1}dk$, so the small-$k$ integrand behaves like $k^{d-1}/k^2=k^{d-3}$:

$${\int }_0\frac{k^{d-1}dk}{|\mathbf{B}|M+c{k}^2}\underset{\mathbf{B}\to 0}{\to }\{\begin{array}{ll}\text{finite},& d\ge 3,\\ {\int }_0\frac{dk}{k}\sim -\frac{1}{2}\ln (|\mathbf{B}|M)\to \infty ,& d=2,\\ {\int }_0\frac{dk}{|\mathbf{B}|M+c{k}^2}\sim (|\mathbf{B}|M{)}^{-1/2}\to \infty ,& d=1.\end{array}$$

In $d=2$ the integral diverges logarithmically in the infrared; in $d=1$ it diverges as a power. Since the left side $S(S+1)$ is finite, the only way the inequality can hold as $\mathbf{B}\to 0$ is $M\to 0$ — and solving the self-consistent bound $S(S+1)\gtrsim M^2{k}_{B}T\int d^{d}k/(|\mathbf{B}|M+c{k}^2)$ for $M$ gives exactly the rates quoted at the top: $M\lesssim 1/\sqrt{T|\ln |\mathbf{B}||}$ in $d=2$ and $M\lesssim (|\mathbf{B}|/T^2{)}^{1/3}$ in $d=1$. The $\int d^{2}k/k^2$ magnon divergence is the whole theorem. Physically: in $d\le 2$ there is so much phase space in the soft, long-wavelength Goldstone modes that exciting them costs almost nothing, and their thermal population washes out the order at any finite $T$. (Equivalently, the real-space transverse spin correlations decay — algebraically in 2D, faster in 1D — rather than saturating to a nonzero value.)

The two essential hypotheses — and the loopholes that restore order

The proof makes crystal-clear which two ingredients are load-bearing, and every route to real 2D magnetism is an attack on one of them.

(1) Continuous symmetry. The $k^2$ in the denominator is the gaplessness of the Goldstone mode that a continuous broken symmetry guarantees (no restoring force for a uniform global rotation). Break the symmetry down to discrete, or otherwise open a gap $\Delta$ in the magnon spectrum, and the integrand becomes $1/(\Delta +c{k}^2)$, which is infrared-finite in 2D — the divergence is cut off and order survives. The two physical gap-openers:

• Magnetic anisotropy. Single-ion anisotropy $D(S^z{)}^2$, anisotropic (Ising-like) exchange, or the antisymmetric/symmetric anisotropic parts of ${\mathbf{J}}_{ij}$ all gap the magnon at $\mathbf{k}=0$. This is precisely why real 2D magnets need anisotropy: an out-of-plane easy axis turns the would-be Goldstone mode into a gapped magnon, the IR integral converges, and a finite Curie temperature appears. The connection runs straight to DMI & anisotropy (the relativistic, SOC-derived terms that supply the gap) and is the physical content of the gapping discussion in LSWT. Mermin–Wagner does not forbid 2D order; it forbids gapless (isotropic) 2D order, and demands the gap be put there by anisotropy.
• Long-range dipolar coupling. The magnetostatic dipole–dipole interaction is long-range — it violates the second-moment / short-range hypothesis — and produces a non-analytic $|\mathbf{k}|$ term in the magnon dispersion that also tames the IR integral. Real thin films have it whether you want it or not.

(2) Short-range interaction. The finite second moment ${\sum }_j|J_{ij}|R_{ij}^2<\infty$ is what makes the small-$k$ stiffness analytic ($\propto k^2$). Sufficiently slowly-decaying exchange, $J(R)\sim R^{-(d+\sigma )}$ with small $\sigma$, gives a sub-$k^2$ dispersion and can stabilise order in 2D (and even 1D) — the dipolar case is the physically important instance.

Contrasts: Ising, Onsager, and BKT

Two neighbouring cases sharpen what the theorem does and does not say:

• Discrete symmetry (2D Ising). The Ising model breaks only a discrete ${\mathbb{Z}}_2$ symmetry, so there is no Goldstone mode and the whole argument is vacuous. Indeed the 2D Ising model does order at finite $T$ — Onsager’s exact solution gives $T_c$ via $\sinh (2J/k_B{T}_c)=1$. So a 2D magnet with strong easy-axis (Ising-like) anisotropy is in the Ising universality class and orders; this is the extreme limit of “anisotropy restores order.”
• 2D $XY$ model and BKT. The $XY$ model has a continuous $O(2)$ symmetry, so Mermin–Wagner applies: $⟨\mathbf{S}⟩=0$ at all $T>0$, no true long-range order. Yet it is not disordered at low $T$: the spin–spin correlations decay algebraically (power-law, $⟨\mathbf{S}(0)\cdot \mathbf{S}(r)⟩\sim r^{-\eta (T)}$) below a transition temperature $T_{\mathrm{BKT}}$ and exponentially above it. This is quasi-long-range order, and the transition is the topological Berezinskii–Kosterlitz–Thouless unbinding of vortex–antivortex pairs — a phase transition with no local order parameter. Mermin–Wagner kills the magnetisation but says nothing about this topological order; the two coexist consistently in 2D.

The summary I keep in mind: gapless + continuous + short-range + $d\le 2$ $\Rightarrow$ no order. Knock out any one — gap it with anisotropy, make it discrete (Ising), make it long-range (dipolar) — and 2D magnetism becomes possible. The experiments on monolayer magnets are the direct vindication: monolayer CrI${}_3$ and bilayer/monolayer CrGeTe${}_3$ (CrSBr) retain ferromagnetism down to a single layer, and the published interpretation attributes this to the magnetic anisotropy opening the spin-wave gap — exactly the Mermin–Wagner loophole — which is why the upstream calculation must pin down the anisotropy and DMI quantitatively, not just the isotropic $J$.

Builds toward: DMI & anisotropy · CrSBr 2D magnets

Key references

• N. D. Mermin & H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133 (1966) — the theorem and the Bogoliubov-inequality proof.
• P. C. Hohenberg, “Existence of long-range order in one and two dimensions,” Phys. Rev. 158, 383 (1967) — the parallel result for superfluid and crystalline order.
• B. Huang et al., “Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit,” Nature 546, 270 (2017) — monolayer CrI${}_3$; anisotropy-stabilised 2D ferromagnetism.
• C. Gong et al., “Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals,” Nature 546, 265 (2017) — few-layer CrGeTe${}_3$; anisotropy as the order-restoring ingredient.